Sure, let's break this problem down step-by-step.
### Volume of the Rubber Ball
First, we need to calculate the volume of a rubber ball. The formula for the volume [tex]\( V \)[/tex] of a sphere is given by:
[tex]\[
V = \frac{4}{3} \pi r^3
\][/tex]
where [tex]\( r \)[/tex] is the radius of the sphere. Given that the radius [tex]\( r = 1.7 \)[/tex] cm, we can plug this into the formula:
[tex]\[
V = \frac{4}{3} \pi (1.7)^3
\][/tex]
Now, calculating [tex]\( 1.7^3 \)[/tex]:
[tex]\[
1.7^3 = 1.7 \times 1.7 \times 1.7 = 4.913
\][/tex]
Next, multiplying by [tex]\( \frac{4}{3} \)[/tex] and [tex]\( \pi \)[/tex]:
[tex]\[
V = \frac{4}{3} \times \pi \times 4.913 \approx 20.58 \, \text{cm}^3
\][/tex]
So, the volume of one rubber ball, rounded to the nearest hundredth, is:
[tex]\[
20.58 \, \text{cm}^3
\][/tex]
### Cost of Producing One Ball
The cost of producing one ball depends on the volume and the cost per cubic centimeter. Given that the cost per cubic centimeter of rubber is \[tex]$0.0045,
we calculate the production cost as follows:
\[
\text{Cost} = \text{Volume} \times \text{Cost per cm}^3
\]
\[
\text{Cost} = 20.58 \, \text{cm}^3 \times 0.0045 \, \text{\$[/tex] / cm}^3
\]
[tex]\[
\text{Cost} = 0.09261
\][/tex]
Rounding this to the nearest cent, the cost of producing one ball is:
[tex]\[
\$0.09
\][/tex]
### Profit Per Ball
Finally, the profit made on each ball can be calculated by subtracting the production cost from the selling price. Given that the company sells a ball for \[tex]$0.50:
\[
\text{Profit} = \text{Selling Price} - \text{Production Cost}
\]
\[
\text{Profit} = 0.50 - 0.09 = 0.41
\]
Therefore, the profit on each ball is:
\[
\$[/tex]0.41
\]
In summary:
- The volume of one rubber ball is [tex]\( 20.58 \, \text{cm}^3 \)[/tex].
- The cost of producing one ball is \[tex]$0.09.
- The profit per ball is \$[/tex]0.41.
